CN108306735A - The hardware implementation method and its system of elliptic curve point multiplication operation - Google Patents
The hardware implementation method and its system of elliptic curve point multiplication operation Download PDFInfo
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- CN108306735A CN108306735A CN201711471815.8A CN201711471815A CN108306735A CN 108306735 A CN108306735 A CN 108306735A CN 201711471815 A CN201711471815 A CN 201711471815A CN 108306735 A CN108306735 A CN 108306735A
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- abscissa
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- curve point
- elliptic curve
- projective coordinates
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3066—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves
Abstract
The invention discloses a kind of hardware implementation methods and its system of elliptic curve point multiplication operation, are related to data hardware encryption technical field.The hardware implementation method of elliptic curve point multiplication operation of the present invention includes the following steps：Obtain the curve point abscissa and point multiplying factor on elliptic curve；According to the curve point abscissa and the coordinate parameters under described multiplying factor setting projective coordinates；The coordinate parameters are calculated to obtain the abscissa under the projective coordinates；The abscissa under the projective coordinates is converted to obtain dot product result.Curve point abscissa on elliptic curve is calculated with point multiplying factor under projective coordinates pattern by the representation method based on projective coordinates, without being pre-processed to input data, is simplified algorithm, saved operation time by technical solution of the present invention.
Description
Technical field
The present invention relates to data hardware encryption technical fields, more particularly to a kind of the hard of elliptic curve point multiplication operation
Part implementation method and its system.
Background technology
In information security field, most important link is exactly information encryption, and encryption technology is again main in the application
In terms of software cryptography and hardware encryption.In existing common encryption method (including symmetric encryption method and asymmetric encryption
Method) in, since the complexity and hardware aspect of algorithm are relatively difficult to achieve, asymmet-ric encryption method is by higher as safety coefficient
Encryption method use.In existing asymmet-ric encryption method, elliptic curve cryptography is safe but relatively difficult to achieve, and ellipse
The speed of point multiplication operation determines the speed of whole cryptographic calculation again in circular curve cryptographic calculation, therefore dot product module can be at last
One of most important module when enciphering and deciphering algorithm hardware realization.
In the elliptic curve cryptography design for being currently based on finite field, elliptic curve is referred to by the Lars Wei Ersite
（Weierstrass）The plane curve that equation determines.In the prior art, the hardware realization of elliptic curve cryptography dot product module is most
Using the conventional point multiplication operation method based on prime field, dimension is also relatively small, generally uses 163,191,233；Simultaneously as often
The algorithm that uses of rule point multiplication operation is more complex, therefore the module that while leading to hardware realization is designed is more, and function is realized cumbersome, and needs
It to pre-process in advance, the plenty of time can be consumed.
Invention content
The main purpose of the present invention is to provide a kind of hardware implementation method and its system of elliptic curve point multiplication operation, purports
In the simplified elliptic hypothesis curve encryption algorithm, save operation time.
To achieve the above object, the present invention provides a kind of hardware implementation method of elliptic curve point multiplication operation, including following
Step：
Obtain the curve point abscissa and point multiplying factor on elliptic curve；
According to the curve point abscissa and the coordinate parameters under described multiplying factor setting projective coordinates；
The coordinate parameters are calculated to obtain the abscissa under the projective coordinates；
The abscissa under the projective coordinates is converted to obtain dot product result.
Preferably, the coordinate parameters being arranged with described multiplying factor according to the curve point abscissa under projective coordinates
Further include：
The coordinate parameters X is set1、Z1、X2、Z2Value：Enable X1Equal to the curve point abscissa Px、X2Equal to Px 4+b、 Z1Deng
In 1, Z2Equal to Px 2, wherein b is parameter preset.
Preferably, the calculating coordinate parameters further include to obtain the abscissa under the projective coordinates：It is described to penetrate
Abscissa under shadow coordinate is calculated by equation x=X/Z；
According to the digit n of the dot product coefficient k to coordinate parameters（X1, Z1）、（X2, Z2）Carry out n-1 point plus fortune successively respectively
Calculation and extraordinarily operation, to obtain the value of X, Z.
Preferably, the abscissa under the conversion projective coordinates further includes to obtain dot product result：Pass through equation Qx=
X/Z calculates dot product result, wherein X and Z is to calculate the fortune exported after the coordinate parameters by point add operation and extraordinarily operation
Calculate result.
Preferably, the curve point abscissa obtained on elliptic curve and point multiplying factor further include：Input the dot product
The value of coefficient k, and it is converted into binary number.
The present invention also provides a kind of elliptic curve point multiplication operation systems, including：
Register for storing the curve point abscissa on elliptic curve and point multiplying factor；
It is connected to the register and obtains the controller of the curve point abscissa and described multiplying factor；
The controller includes computing module, the dot product result for calculating the curve point abscissa and described multiplying factor.
Preferably, the controller further includes conversion module, for described multiplying factor to be converted to binary number, and
Curve point abscissa is converted into the coordinate parameters under projective coordinates.
Preferably, the computing module includes point add operation module, extraordinarily computing module and coordinate transferring；
The point add operation module and the extraordinarily computing module obtain described multiplying factor and the coordinate parameters are counted
It calculates；
The point add operation module and the extraordinarily computing module by the result after calculating be sent to the coordinate transferring with
Obtain dot product result.
Preferably, the point add operation module and the extraordinarily computing module are the multiplier module in finite field.
Preferably, the coordinate transferring is inversion calculation module.
Technical solution of the present invention by the representation method based on projective coordinates, by elliptic curve curve point abscissa and
Point multiplying factor is calculated under projective coordinates pattern, without being pre-processed to input data, simultaneously because only with point
Abscissa does point multiplication operation, so simplifying algorithm, having saved operation time.
Description of the drawings
Fig. 1 is the hardware implementation method flow diagram of elliptic curve point multiplication operation of the present invention；
Fig. 2 is elliptic curve point multiplication operation system principle schematic diagram of the present invention.
The embodiments will be further described with reference to the accompanying drawings for the realization, the function and the advantages of the object of the present invention.
Specific implementation mode
It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not intended to limit the present invention.
The following further describes the present invention with reference to the drawings.
The present invention is to be based on Montgomery（Montgomery）Algorithm for Scalar Multiplication, in the algorithm, the representation method of point abandons biography
System（x,y）Representation, uses the representation method of projective coordinates pattern (X, Z) instead, and design module is reduced in hardware realization,
To reduce the occupancy of resource.The hardware implementation method and its system of elliptic curve point multiplication operation of the present invention, which use, is characterized as 2
Finite field gf (2n) implementation of the elliptic curve as hardware, wherein dimension n=283, compared with the prior art in use
Dimension 163,191,233, encryption safe coefficient can be greatly improved,
As shown in Figure 1, the present embodiment provides a kind of hardware implementation method of elliptic curve point multiplication operation, include the following steps：
Obtain the curve point abscissa and point multiplying factor on elliptic curve；According to the curve point abscissa and described multiplying factor
Coordinate parameters under projective coordinates are set；The coordinate parameters are calculated to obtain the abscissa under the projective coordinates；Conversion institute
The abscissa under projective coordinates is stated to obtain dot product result.
In elliptic curve encryption processes, elliptic curve formula is：y2 + xy = x3 + ax2+ b, ciphering process are
To seek elliptic curve dot product Q=k*Px=Qx, wherein PxValue be elliptic curve on any point（x,y）Abscissa x value；k
For user, customized parameter, expression need to do the number of point multiplication operation before encryption, and what is be generally randomly generated is not 0 number
Word；A and b is preset parameter, and during actual operation, general value is 1, and the speed of such point multiplication operation is up to most fast.
The coordinate parameters X is set1、Z1、X2、Z2Value：Enable X1Equal to the curve point abscissa Px、X2Equal to Px 4+b、
Z1Equal to 1, Z2Equal to Px 2, wherein b is parameter preset.The value of the dot product coefficient k is inputted, and is converted into binary number.
Integer k is converted into the binary number of computer capacity identification；By elliptic curve point abscissa PxIt is converted into projective coordinates（X, Z）Mould
Formula.
Preferably, the calculating coordinate parameters further include to obtain the abscissa under the projective coordinates：It is described to penetrate
Abscissa under shadow coordinate is calculated by equation x=X/Z；According to the digit n of the dot product coefficient k to coordinate parameters（X1, Z1）、
（X2, Z2）N-1 point add operation and extraordinarily operation are carried out successively respectively, to obtain the value of X, Z.It calculates binary number k and projection is sat
The point multiplication operation of (X, Z) is marked as a result, with the abscissa x of another point on elliptic curve under projective coordinates pattern after being multiplied.
Preferably, the abscissa under the conversion projective coordinates further includes to obtain dot product result：Pass through equation Qx=
X/Z calculates dot product result, wherein X and Z is to calculate the fortune exported after the coordinate parameters by point add operation and extraordinarily operation
Calculate result.Another point abscissa x under the projective coordinates pattern being calculated is converted into the horizontal seat of the point under affine coordinate pattern
Mark QxTo get to the result of dot product.
The point multiplication operation process of the present embodiment is as follows：
S1, judge k and PxWhether it is 0, Q is exported if 0x=0 and terminate；
S2, integer k is converted into binary number（kn-1kn-2…k1k0）, n indicates the digit of binary k；
S3, the setting coordinate parameters X1、Z1、X2、Z2Value：Enable X1=Px, X2=Px 4+ b, Z1=1、Z2=Px 2, wherein b is 1；Wherein
X1、Z1、X2、Z2For the point P on elliptic curvexValue under projective coordinates；
S4, since a secondary high position of k judge that its value carries out different point add operations and extraordinarily operation respectively for 1 or 0 successively, in terms of
Calculate the abscissa x under projective coordinates（X, Z）, i.e., successively from kn-2To k0Judge that the value of this be 1 is still 0：
Work as kiWhen=1 (value range of i is n-2 to 0), pass through Madd (X1, Z1, X2, Z2), Mdouble (X2, Z2) calculate
X values；Wherein Madd and Mdouble indicates the point add operation under projective coordinates and extraordinarily operation respectively；
Work as kiWhen=0 (value range of i is n-2 to 0), pass through Madd (X2, Z2, X1, Z1), Mdouble (X1, Z1) calculate
X values；
Specifically, work as kiWhen=1, Madd (X1, Z1, X2, Z2) by etc. Formula X=(X1*Z2 + X2*Z1) 2, Z=x*Z3 +
( X1*Z2)*( X2*Z1) X, Z is calculated；Enable X1=X, Z1=Z such as brings at the Formula X=X2 4 + b·Z2 4, Z=Z2 2·X2 2In terms of
Calculate Mdouble (X2, Z2), then assign X, Z for being calculated to the X in subsequent cycle calculating respectively2、Z2；
As n=0, preserve result of calculation X, Z to get to the abscissa x under projective coordinates=（X, Z）；
S5, output point multiplication operation result of calculation Qx=x=X/Z。
In embodiments of the present invention, the ordinate of P points and it is not involved in Algorithm for Scalar Multiplication calculating process, is not influencing encryption and decryption mistake
Under the premise of journey, the time that operation is consumed is greatly reduced, also saves the hardware resource that Algorithm for Scalar Multiplication is related to.
Technical solution of the present invention is horizontal by the curve point on the elliptic curve of input by the representation method based on projective coordinates
Coordinate and point multiplying factor are calculated under projective coordinates pattern, without being pre-processed to input data, are simplified algorithm, are saved
Operation time.
As shown in Fig. 2, the present invention also provides a kind of elliptic curve point multiplication operation systems, including：
Register for storing the curve point abscissa on elliptic curve and point multiplying factor；It is connected to the register and obtains
The controller of the curve point abscissa and described multiplying factor；The controller includes computing module, for calculating the song
The dot product result of line point abscissa and described multiplying factor.
Preferably, the controller further includes conversion module, for described multiplying factor to be converted to binary number, and
Curve point abscissa is converted into the coordinate parameters under projective coordinates.In some embodiments, dot product coefficient k is in input register
When can be automatically converted to binary number, then do not need additional conversion module and converted.
Preferably, the computing module includes point add operation module, extraordinarily computing module and coordinate transferring；The point
Add that computing module and the extraordinarily computing module obtain described multiplying factor and the coordinate parameters are calculated；The point adds
Result after calculating is sent to the coordinate transferring to obtain dot product result by computing module and the extraordinarily computing module.
Preferably, the point add operation module and the extraordinarily computing module are the multiplier module in finite field.The seat
Mark conversion module is inversion calculation module.
According to above-described embodiment it is found that the calculative equation of computing module includes point add operation and extraordinarily operation, and point
Operation and extraordinarily operation is added mainly to realize that in a particular embodiment, computing module then only needs multiplier module by multiplication, addition
It can be realized：Addition in point add operation and extraordinarily operation can be can be realized by simple exclusive or, need not in addition be increased and be added
Method module；Multiplication in point add operation and extraordinarily operation calls multiplier module respectively by control module.And it completes point and adds
After extraordinarily X, Z is calculated in operation, the abscissa under projective coordinates is converted to by conversion module under radial pattern for operation
Coordinate calculates Qx=X/Z also needs that dot product result is calculated using inversion calculation module.The present invention is by less module
The point multiplication operation in elliptic curve encryption algorithm can be achieved, reduce port number when hardware realization, increase arithmetic speed.
In further embodiments, above-mentioned multiplier module includes the first multiplier module and the second multiplier module, on making
Stating the ad eundem multiplying in embodiment in equation can be carried out at the same time, and can increase multiplier module, to accelerate point multiplication operation
Speed.
It should be understood that these are only the preferred embodiment of the present invention, the scope of the claims of the present invention cannot be therefore limited,
It is every to utilize equivalent structure or equivalent flow shift made by description of the invention and accompanying drawing content, directly or indirectly use
In other related technical areas, it is included within the scope of the present invention.
Claims (10)
1. a kind of hardware implementation method of elliptic curve point multiplication operation, which is characterized in that include the following steps：
Obtain the curve point abscissa and point multiplying factor on elliptic curve；
According to the curve point abscissa and the coordinate parameters under described multiplying factor setting projective coordinates；
The coordinate parameters are calculated to obtain the abscissa under the projective coordinates；
The abscissa under the projective coordinates is converted to obtain dot product result.
2. the hardware implementation method of elliptic curve point multiplication operation according to claim 1, which is characterized in that described according to institute
Stating the coordinate parameters that curve point abscissa is arranged with described multiplying factor under projective coordinates further includes：
The coordinate parameters X is set1、Z1、X2、Z2Value：Enable X1Equal to the curve point abscissa Px、X2Equal to Px 4+b、 Z1Deng
In 1, Z2Equal to Px 2, wherein b is parameter preset.
3. the hardware implementation method of elliptic curve point multiplication operation according to claim 2, which is characterized in that the calculating institute
State coordinate parameters further includes to obtain the abscissa under the projective coordinates：Abscissa under the projective coordinates by x=（X,
Z）It indicates；
According to the digit n of the dot product coefficient k to coordinate parameters（X1, Z1）、（X2, Z2）Carry out n-1 point add operation successively respectively
Extraordinarily operation, to obtain the value of X, Z.
4. the hardware implementation method of elliptic curve point multiplication operation according to claim 2, which is characterized in that the conversion institute
The abscissa under projective coordinates, which is stated, to obtain dot product result further includes：Pass through equation Qx=X/Z calculate dot product result, wherein X and
Z is to calculate the operation result exported after the coordinate parameters by point add operation and extraordinarily operation.
5. the hardware implementation method of elliptic curve point multiplication operation according to claim 2, which is characterized in that the acquisition is ellipse
Curve point abscissa on circular curve and point multiplying factor further include：
The value of the dot product coefficient k is inputted, and is converted into binary number.
6. a kind of elliptic curve point multiplication operation system, which is characterized in that including：
Register for storing the curve point abscissa on elliptic curve and point multiplying factor；
It is connected to the register and obtains the controller of the curve point abscissa and described multiplying factor；
The controller includes computing module, the dot product result for calculating the curve point abscissa and described multiplying factor.
7. elliptic curve point multiplication operation system according to claim 6, which is characterized in that the controller further includes conversion
Curve point abscissa for described multiplying factor to be converted to binary number, and is converted to the seat under projective coordinates by module
Mark parameter.
8. elliptic curve point multiplication operation system according to claim 7, which is characterized in that the computing module includes that point adds
Computing module, extraordinarily computing module and coordinate transferring；
The point add operation module and the extraordinarily computing module obtain described multiplying factor and the coordinate parameters are counted
It calculates；
The point add operation module and the extraordinarily computing module by the result after calculating be sent to the coordinate transferring with
Obtain dot product result.
9. elliptic curve point multiplication operation system according to claim 8, which is characterized in that the point add operation module and institute
It is the multiplier module in finite field to state extraordinarily computing module.
10. elliptic curve point multiplication operation system according to claim 8, which is characterized in that the coordinate transferring is
Inversion calculation module.
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| CN112099760A (en) * | 2020-08-24 | 2020-12-18 | 清华大学 | Single multiplier seamless scheduling method for point addition and point doubling in SM2 cryptographic algorithm |
| CN113691543A (en) * | 2021-08-25 | 2021-11-23 | 苏州国芯科技股份有限公司 | Data encryption method and device based on elliptic curve, computer equipment and medium |
| CN114001650A (en) * | 2021-09-16 | 2022-02-01 | 北京市测绘设计研究院 | Method for encrypting conversion parameters of earth coordinate system and arbitrary plane coordinate system |
| CN114489571A (en) * | 2022-04-15 | 2022-05-13 | 广州万协通信息技术有限公司 | Asymmetric algorithm calculation circuit |
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| CN115062565A (en) * | 2022-06-22 | 2022-09-16 | 北京理工大学 | Design method of low-delay elliptic curve point multiplication circuit |
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Cited By (12)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109889341A (en) * | 2019-01-15 | 2019-06-14 | 思力科（深圳）电子科技有限公司 | Data processing method, electronic tag and radio-frequency card reader |
| CN111740820A (en) * | 2020-08-10 | 2020-10-02 | 灵长智能科技(杭州)有限公司 | ECC-based finite field operation scheduling method, device, equipment, medium and system |
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| CN112099760B (en) * | 2020-08-24 | 2022-11-11 | 清华大学 | Single multiplier seamless scheduling method for point addition and doubling in SM2 cryptographic algorithm |
| CN113691543A (en) * | 2021-08-25 | 2021-11-23 | 苏州国芯科技股份有限公司 | Data encryption method and device based on elliptic curve, computer equipment and medium |
| CN114001650A (en) * | 2021-09-16 | 2022-02-01 | 北京市测绘设计研究院 | Method for encrypting conversion parameters of earth coordinate system and arbitrary plane coordinate system |
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| CN114489571A (en) * | 2022-04-15 | 2022-05-13 | 广州万协通信息技术有限公司 | Asymmetric algorithm calculation circuit |
| CN114531241A (en) * | 2022-04-22 | 2022-05-24 | 北京智芯微电子科技有限公司 | Data encryption method and device, electronic equipment using data encryption method and storage medium |
| CN114531241B (en) * | 2022-04-22 | 2022-08-30 | 北京智芯微电子科技有限公司 | Data encryption method and device, electronic equipment using data encryption method and storage medium |
| CN115062565A (en) * | 2022-06-22 | 2022-09-16 | 北京理工大学 | Design method of low-delay elliptic curve point multiplication circuit |
| CN115062565B (en) * | 2022-06-22 | 2024-01-05 | 北京理工大学 | Low-delay elliptic curve point multiplication circuit design method |
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Application publication date: 20180720 |